This is something I thought about today but have no idea how to approach.
We are given a right circular cone with lateral length L and angle at the base $\alpha$. A curve along the surface of the cone is connecting the base with the apex. The slope of the curve at each point (i.e. the angle between the tangent at each point and the plane of the base of the cone) is not bigger than $\beta<\alpha$. What is the minimum length of such a curve?
My intuition says the curve of minimum length is some helix but I have no idea how to prove this. Any pointers to relevant materials are also appreciated.
The trick here is not to attempt to derive the shape of the curve explicitly. (That is possible and arguably not even difficult, but is a bit messy and not really enlightening).
Just assume that we have some parameterization $\gamma(t)$ and further assume that it's parameterized by arc length.
Let $a(t)$ be the distance between $\gamma(t)$ and the apex.
By purely local geometric reasoning you can now find an inequality for $\frac{da}{dt}$ -- it turns out to be limited by a constant that depends only on $\alpha$ and $\beta$. Now you can figure out what the quickest way for $a$ to drop from $L$ to $0$ is.